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MiS Preprint
53/2017
Connecting UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ with partial Hadamard matrices
Yan-Ling Wang, Mao-Sheng Li, Shao-Ming Fei and Zhu-Jun Zheng
Abstract
We study the unextendible maximally entangled bases (UMEB) in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ and connect the problem to the partial Hadamard matrices. We show that for a given special UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$, there is a partial Hadamard matrix which can not be extended to a complete Hadamard matrix in $\mathbb{C}^{d}$. As a corollary, any $(d-1)\times d$ partial Hadamard matrix can be extended to a complete Hadamard matrix, which answers a conjecture about $d=5$. We obtain that for any $d$ there is a UMEB except for $d=p\ \text{or}\ 2p$, where $p\equiv 3\mod 4$ and $p$ is a prime. The existence of different kinds of constructions of UMEBs in $\mathbb{C}^{nd}\bigotimes\mathbb{C}^{nd}$ for any $n\in \mathbb{N}$ and $d=3\times5 \times7$ is also discussed.
